F-Sin (theta) lens system and method of use of same

ABSTRACT

An optical system for focusing an incident light so as to obtain a linear output (as a function of the wavelength, or of the order of diffraction, or of the grating spacial frequency or of the inverse of the index of refraction) comprises a diffraction grating and a lens, or a group of lenses, having a f-Sin(θ) characteristic, where f is the effective focal length of the lens, or group of lenses, and θ is an output angle of the light exiting from the diffraction grating. Therefore, a light incident on the diffraction grating reaches the lens, or the group of lenses, and results in a linear output. Using the present invention in a spectrometer reduces the time consuming calibration process required and renders it easier because of the linear output between wavelength and pixels positions. The lens, or group of lenses, has a diffractive, refractive or reflective property, or any combination thereof. The incident light can be a collimated, converging or diverging beam. The diffraction grating can be a plane, concave or convex grating.

BACKGROUND OF THE INVENTION

[0001] 1. Field of the Invention

[0002] The present invention relates to lenses used in optical systemsand, more particularly to a new lens or a new lens group.

[0003] 2. Description of the Prior Art

[0004] Introducing the desired amount of distortion in an optical systemis possible and relatively straight forward for optical designers. Knownfθ lenses (as in U.S. Pat. No. 4,695,132 issued on Sep. 22, 1987 toSakuma and U.S. Pat. No. 4,770,517 issued on Sep. 13, 1998 also toSakuma) used in laser scanner systems are one of the most popular lensesof this type. Over the past decade or two, system improvements haveshifted product emphasis toward electronic features and digitalprocessing, such that optical system design has frequently beenrelegated to a low priority. Recently, there has been a revival ofinterest in solving optical problems with innovative optical design. Onthe other hand, it is possible to simplify considerably the task of thesoftware by designing more sophisticated optical elements.

[0005] For example, a spectrometer uses a diffraction grating to spreadthe spectral component of a light incident of the grating on a one (1)dimensional detector. However, on the detector, the spectral componentsare not spread linearly such that the resolution is thus not constantand the calibration process must be applied very carefully to preventerror.

[0006] It would thus be desirable to have a lens, or group of lens,which when used with a diffraction grating would yield a linear outputthereby substantially eliminating a time consuming calibration process.

OBJECTS OF THE INVENTION

[0007] It is therefore an aim of the present invention to provide a newlens or lens group.

[0008] It is a also an aim of the present invention to provide a newsystem which combines the new lens and a diffraction grating providing alinear output plane as a function of the wavelength (λ), or of the orderof diffraction (m), or of the grating spatial frequency (1/Λ), or of theinverse of the index of refraction (n).

[0009] It is a further aim of the present invention to provide a newlens or lens group with proper amount of distortion to provide af-Sin(θ) characteristic. The optical lens may have a positive or anegative power and it can be refractive, diffractive and reflective orin combination.

SUMMARY OF THE INVENTION

[0010] Therefore, in accordance with the present invention, there isprovided an optical system for obtaining a linear output from anincident light, comprising a diffraction grating and a lens having af-Sin(θ) characteristic, where f is the effective focal length of saidlens and θ is an output angle of the light exiting from said diffractiongrating, wherein a light incident on said diffraction grating reachessaid lens and results in a linear output.

[0011] Also in accordance with the present invention there is providedan optical system for obtaining a linear output from an incident light,comprising a diffraction grating and a group of lenses having a f-Sin(θ)characteristic, where f is the effective focal length of said group oflenses and θ is an output angle of the light exiting from saiddiffraction grating, wherein a light incident on said diffractiongrating reaches said group of lenses and results in a linear output.

[0012] If a f-Sin(θ) lens or lens group is introduced for instance in aspectrometer, this new lens provides a correction for the deflection ofthe laser beam which takes place at a linear position with thewavelength in the detector plane. Specifically, when the diffractiongrating is positioned on the entrance pupil of the f-Sin(θ) lens havingan effective focal length of f, with respect to the optical axisthereof, the beam will be focussed onto the detector plane at a pointwhich is displaced by a distance of f-Sin(θ) from the optical axis.According with the grating equation at normal incidence${{{Sin}(\theta)} = \frac{m \cdot \lambda}{n \cdot \Lambda}},$

[0013] where θ is the diffraction angle, m is the diffraction order, λis the wavelength, n the index of the refraction and Λ the gratingperiod, the distance from the optical axis of the focused beam is alinear function of the wavelength. Then the calibration can besimplified because the spectral component of a light signal spread onthe detector is linearly distributed on the linear detector. Furthermorethe resolution is constant over the wavelength operating range. Thedesignation “f-Sin(θ) lens”, or “f-λ lens”, is derived from such fact.

[0014] The present invention can solve the problem associated with thenonlinear imaging process in the spectrometer and it can solve alsoother problems. This invention is intended to provide a new lens or lensgroup which can be used with a diffraction grating to provide a linearoutput plane with the wavelength (λ), but also with the order ofdiffraction (m), or with the grating spatial frequency (1/Λ), or withthe inverse of the index of refraction (n).

[0015] Accordingly, the invention provides a new lens or a new lensgroup with proper amount of distortion to provide a f-Sin(θ)characteristic. The optical lens or lens group of the present inventionhas a positive or a negative power and it can be refractive, diffractiveand reflective or a combination of all these properties.

BRIEF DESCRIPTION OF THE DRAWINGS

[0016] Having thus generally described the nature of the invention,reference will now be made to the accompanying drawings, showing by wayof illustration a preferred embodiment thereof, and in which:

[0017]FIG. 1 is the optical layout of the refractive configuration of af-Sin(θ) single lens in accordance with the present invention;

[0018]FIG. 2 graphically illustrates the f-Sin(θ) correction in thearrangement of FIG. 1;

[0019]FIG. 3 is the optical layout of the refractive configuration of af-Sin(θ) doublet also in accordance with the present invention;

[0020]FIG. 4 graphically illustrates the f-Sin(θ) correction in thearrangement of FIG. 3;

[0021]FIG. 5 is the optical layout illustration of a reflectiveconfiguration of the f-Sin(θ) lens further in accordance with thepresent invention;

[0022]FIG. 6 graphically illustrates the f-Sin(θ) correction in thearrangement of FIG. 5; and

[0023]FIG. 7 is a schematic view of the diffraction grating and the lenswith the diffraction angle and the incident angle are different.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0024] The f-Sin(θ) lens system of the present invention represents inits most simple arrangement a single lens L, as shown in FIG. 1. Asshown, a radius of curvature of a first surface is represented by R1, aradius of curvature of a second opposed surface by R2, the separationbetween the lens surfaces R1 and R2 by d, and the index of refraction ofthe lens L is represented by N. The variables described above form thelens L. The lens variables are calculated to minimize the f-Sin(θ)characteristic. This can be done by a proper merit function in anyoptical design software.

[0025] The f-Sin(θ) characteristic can be defined by${{\lbrack {f \cdot {{Sin}(\theta)}} \rbrack \text{characteristic}} = {\frac{{H(\theta)} - {f \cdot {{Sin}(\theta)}}}{f \cdot {{Sin}(\theta)}} \cdot 100}},$

[0026] where θ represents the diffraction angle from the diffractiongrating and H(θ) corresponds to the height of the focused beam in theimage plane with respect to the optical axis of the focused beam [showthe optical axis on the drawings]. The f-Sin(θ) characteristic signifiesa deviation from an ideal f-Sin(θ) characteristic, in percentage. For anideal f-Sin(θ) lens or lens group, H(θ)=fSin(θ) for every angle ofincidence, and the f-Sin(θ) characteristic is equal to zero.

[0027] The permissible values of f-Sin(θ) characteristic are difficultto evaluate, it may depend on the kind of application. It is relativelyeasy to use +/−0.1% as a good starting point. This value is a typicalnumber used for the fθ lens. FIG. 2 shows the f-Sin(θ) characteristic ofthe lens of FIG. 1.

[0028] It is the second object of the present invention to provide a newoptical system S. This new system combines the f-Sin(θ) lens and adiffraction grating. The main optical elements of the new optical systemS comprise of a diffraction grating G placed on the entrance pupil E ofthe f-Sin(θ) lens (see on FIG. 1), the f-Sin(θ) lens L and an imageplane P which in the illustrated embodiments takes the form of a focalplane (in other cases, the image plane could be a conjugated plane). Theoptical system S of the present invention operates by transmitting alight beam, for instance a collimated light beam, from a light sourcethrough the diffraction grating G, the diffracted light passes throughthe f-Sin(θ) lens L and is focused in the image plane P.

[0029] If the light incident on the grating G is polychromatic, thediffraction grating G spreads the light in the “m” order of diffraction(m=1, by example). Then, the position of the light spread in the imageplane P is directly proportional to the wavelength of the light. We canderive this fact from the grating equation:${H(\theta)} = {{f \cdot {{Sin}(\theta)}} = {{f \cdot \frac{m \cdot \lambda}{n \cdot \Lambda}} = {{A \cdot \lambda} = {{H(\lambda)}.}}}}$

[0030] As we can see, the height H(θ) is a linear function of thevariable λ (i.e. the light's wavelength). To respect the linearity, theorder of diffraction (m), the index (n) and the grating period (Λ) mustbe constant.

[0031] The designation “f-λ lens” is derived from such fact.

[0032] If the light incident on the grating G is monochromatic, thediffraction grating G diffracts the laser beam into several diffractionorders. The position of the diffraction order in the image plane P isthen directly proportional to the diffraction order produced by thediffraction grating following the equation:${H(\theta)} = {{f \cdot {{Sin}(\theta)}} = {{f \cdot \frac{m \cdot \lambda}{n \cdot \Lambda}} = {{B \cdot m} = {{H(m)}.}}}}$

[0033] As we can see, the height H(θ) is a linear function of thevariable m (that is the order of diffraction). To respect the linearity,the wavelength (λ), the index (n) and the grating period (Λ) must beconstant.

[0034] If the light incident on the grating G is monochromatic, thediffraction grating diffracts the laser beam into one diffraction order.The position of the diffracted light in the image plane P is thendirectly proportional the spatial frequency of the diffraction gratingfollowing the equation:${H(\theta)} = {{{f \cdot {Sin}}\quad (\theta)} = {{f \cdot \frac{m \cdot \lambda}{n \cdot \Lambda}} = {{C \cdot \frac{1}{\Lambda}} = {{H( \frac{1}{\Lambda} )}.}}}}$

[0035] As we can see, the height H(θ) is a linear function of thevariable 1/Λ (that is the reverse of the grating period). To respect thelinearity, the wavelength (λ), the index (n) and the diffraction order(m) must be constant.

[0036] If the light is not normal incident but made an angle β with thediffraction grating normal GN (see FIG. 7) on the diffraction grating G,the linearity is respected as the following equation: $\begin{matrix}{{H(\theta)} = {{f \cdot {{Sin}(\theta)}} = \quad {f \cdot ( {\frac{m \cdot \lambda}{n \cdot \Lambda} + {{Sin}(\beta)}} )}}} \\{= \quad {{A\quad \lambda} + E}} \\{= \quad {{Bm} + E}} \\{= \quad {{C\frac{1}{\Lambda}} + E}}\end{matrix}.$

[0037] where E is a new constant equal to the following equation:

E=f·Sin(β)

[0038] The diffraction grating may be plane, convex or concave, andother possible gratings may be used, for instance an asphericdiffraction grating.

EXAMPLES OF PREFERRED EMBODIMENTS

[0039] All optical surfaces of the present invention respect thefollowing sag equation:${{z(r)} = \frac{c \cdot r^{2}}{1 + \sqrt{1 - {( {1 + \delta} ) \cdot c^{2} \cdot r^{2}}}}},$

[0040] where c is the curvature (the reciprocal of the radius ofcurvature (1/R)), r is the radial coordinate and δ is the conicconstant.

[0041]FIG. 1 shows the single f-Sin(θ) lens with the followingprescription: R1 = 73.28 mm d = 2.7 mm Material: BK7 F/# = 100 R2 =−173.16 mm T = 27 mm EFL = 100 mm

[0042] where F/# is the effective focal length divided beam diameter

[0043] where EFL is the effective focal length.

[0044] Design wavelengths 486.1 nm 587.6 nm(primary) 656.3 nm

[0045] The corresponding f-Sin(θ) characteristic is shown in FIG. 2.

[0046]FIG. 3 shows the two air space element f-Sin(θ) doublet lens withthe following prescription: R1 = 50.05 mm d1 = 5 mm N1 = 1.5168 F/# 40R2 = −65.24 mm t = 5 mm T = 19.6 mm R3 = −21.10 mm d2 = 3.55 mm N2 =1.7847 EFL = 100 mm R4 = −28.86 mm

[0047] Design wavelengths 486.1 nm 587.6 nm(primary) 656.3 nm.

[0048] In FIG. 3, the F-Sin(O) doublet comprises a first proximal lensL′ having opposed surfaces R1 and R2, and a second distal lens L′ havingopposed surfaces R3 and R4, respectively separated by distances d1 andd2 and having respective indexes of refraction N1 and N2.

[0049] The corresponding f-Sin(θ) characteristic is shown in FIG. 4. Thelenses L′ and L″ are separated by distance t.

[0050]FIG. 5 shows a reflective configuration of a f-Sin(θ) lens L′″.This embodiment can be used when a large bandwidth is required invarious applications. The optical prescriptions are: R1 = −200 mm T =100 mm F/# = 40 δ = 0.908 mm EFL = 100 mm

[0051] Design wavelengths 486.1 nm 587.6 nm (primary) 656.3 nm

[0052] In FIG. 5, the lens L′″ comprises a reflective surface R1 with aconical constant 8.

[0053] The corresponding f-Sin(θ) characteristic is shown in FIG. 6.

[0054] The invention is also valid when the normal of the grating (GN)make an angle α with the optical axis of the f-Sin(θ) lens or lens groupas we can see in FIG. 7. However in this particular case, thediffraction angle θ is not equal to the incident angle on the f-Sin(θ)lens. The definition of H(θ) is the same as presented above but for theincident angle on the lens, we can express the relation as:

H(θ)=H(θ′)=f·Sin(θ′+α)

[0055] where θ′ is the incident angle on the lens respecting the opticalaxis (OA). The angles are positive as shown in the FIG. 7.

[0056] A distance δ between the optical axis of the lens and the centerof the diffraction grating is also possible. In such a case, theinvention is also valid.

1. An optical system for obtaining a linear output from an incidentlight, comprising a diffraction grating and a lens having a f-Sin(θ)characteristic, where f is the effective focal length of said lens and θis an output angle of the light exiting from said diffraction grating,wherein a light incident on said diffraction grating reaches said lensand results in a linear output.
 2. An optical system as defined in claim1, wherein said lens has at least one of diffractive, refractive andreflective properties.
 3. An optical system as defined in claim 1,wherein said incident light is one of a collimated beam, a convergingbeam and a diverging beam.
 4. An optical system as defined in claim 1,wherein said diffraction grating is one of a plane grating, a concavegrating and a convex grating.
 5. An optical system for obtaining alinear output from an incident light, comprising a diffraction gratingand a group of lenses having a f-Sin(θ) characteristic, where f is theeffective focal length of said group of lenses and θ is an output angleof the light exiting from said diffraction grating, wherein a lightincident on said diffraction grating reaches said group of lenses andresults in a linear output.
 6. An optical system as defined in claim 5,wherein said group of lenses has at least one of diffractive, refractiveand reflective properties.
 7. An optical system as defined in claim 5,wherein said incident light is one of a collimated beam, a convergingbeam and a diverging beam.
 8. An optical system as defined in claim 5,wherein said diffraction grating is one of a plane grating, a concavegrating and a convex grating.